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Motivation Estimate the distribution function of the incubation period of HIV/AIDS: –Nonparametrically –Based on censored data: Time of HIV infection is interval censored Time of onset of AIDS is interval censored or right censored

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Computation of the MLE Reduction step: determine the maximal intersections Optimization step: determine the amounts of mass assigned to the maximal intersections

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Computation of the MLE Reduction step: determine the maximal intersections Optimization step: determine the amounts of mass assigned to the maximal intersections

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Existing reduction algorithms Betensky and Finkelstein (1999, Stat. in Medicine) Gentleman and Vandal (2001, JCGS) Song (2001, Ph.D. thesis) Bogaerts and Lesaffre (2003, Tech. report) The first three algorithms are very slow, the last algorithm is of complexity O(n 3 ).

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X = time of HIV infection Y = time of onset of AIDS Z = Y-X = incubation period An example of a parameter we can estimate consis- tently is:

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Conclusions (1) Our algorithms for the parameter reduction step are significantly faster than other existing algorithms. We proved that in general the naive MLE is an inconsistent estimator for our AIDS model.

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Conclusions (2) We explored several methods to repair the inconsistency of the naive MLE. cannot be estimated consistently without additional assumptions. An alternative parameter that we can estimate consistently is:.